**** knows, that's the thing, my laptop doesnt have a touch screen yet they decided to sell it with an operating system designed for a touch screen. Bunch of bellends. I asked them if they could just downgrade me to windows 7 and they said they couldnt.

For HD video, you'll need a dedicated GPU and nothing less than an i5 in my opinion.
Some of the Adobe software requires a 1GB GPU.
The more RAM the better (absolutely minimum 8GB imo).
I'm also really enjoying an SSD.

The problem with HD video is that you will always want more - you'll always want your previews to be higher res, your rendering time shorter, your UI more instant, etc, etc.

I'm quite happy with W8 - just ignore the start screen. Everything else is very nippy and clean. W8.1 is easier to boot straight to desktop. You just get used to doing things is a slightly different (but quicker) way.

Have to say I 'sorted' (loaded classic shell & other software) onto a Asus touchscreen laptop and was quite impressed, it does seem though that Asus have loaded there own 'gesture' software onto it though (the other makes in comet didn't seem to accept them), but yes windows 8 on a machine without a touchscreen is a really quite stupid thing to do (cheers MS for pushing it out).
Wasn't really impressed by the performance considering the i3, 6Gb of ram it was running, would be tempted to say (really need more research tbf) get an i5 at least.

It's not too bad - the HD4000 is the GPU I've got.
There are some photoshop and premier features grayed out (3D, Lighting effects), Speedgrade wont run at all. But After Effects ran without issues.
It would definitely do the job, but can be bettered.
Obviously if that's right at the top end of your budget, then perhaps that might be the best you can hope for.

Also, it keeps telling me I should upgrade to windows 8.1, is this any bettER?

The only reason Windows 8.1 exists is to help people who didn't like 8, so yes, it should work for you.

I was annoyed with 8 until I
a) Setup my 'default apps' so that nothing opens with rubbish metro apps.
b) Realised that pressing the windows key and typing the start of what you want is actually quite a quick way of doing things
c) Decided to completely ignore the 'tile' interface.
d) put 'desktop' as the top-left tile so that I can just hit return after boot to get to the desktop.
e) Got rid of the login screen.

E.g. what are you doing with MATLAB? I have a laptop for matlab use (for my job) and it's not a cheap one! But you'll have different (almost certainly lesser) requirements. But depending on what you want to do with it, good linux support for the laptop might be on your list. If you're just using "standard" matlab and not dicking about with too much mex action then you're probably fine with any OS though.

If you're on student matlab, you're limited to 32bit IIRC so more than 4Gb ram doesn't help you there.

Similarly, how much are you actually editing HD video, what software etc? No point blowing all your money on a discrete GPU if you can't afford the software to take advantage of it, or if it just speeds up a job you only actually do twice a year and could go and grab a coffee whilst it's processing etc.

FWIW I quite like win8 with touchscreens, though I've only used other people's machines as my work stuff's all win7 or ubuntu and my home kit is OSX or ubuntu.

Take a look at the dell outlet store though (remember to add VAT to prices!) both the home one and the business store - the business laptops tend to be better made. At the higher end at least, can't say I have much experience with low end ones.

You said "cheap" but what's your budget? Pretty much anything will do what you want, but might not be brilliant at it. But then clearly something to be brilliant at HD video editing and matlab is out your budget.

E.g. quick look on dell outlet, about £700 all in gets you 8Gb, quad core i7 up to 3.2Ghz, 2Gb 640m GPU, 1080p screen and a 750Gb HD with 32Gb mSATA SSD cache.

At the moment pretty basic stuff with MATLAB, not too sure on editing software, I'd probably end up using GoPro Studio, just for a few edits really.
It need to be able to render at a usable speed though, and play back high resolution footage no problem (something my laptop struggles with).
Budget is not really more than £550 but preferably cheaper.

my laptop doesnt have a touch screen yet they decided to sell it with an operating system designed for a touch screen

You do know that you can use Windows 8 perfectly easily with a keyboard and mouse? I assume your Lenovo has these? Windows 8 has been enhanced to include touch but it is not exclusively a touch OS. It's really not very hard.

Now, whether you like Windows 8 or not is a personal preference and you may not like it. But to describe it as shitty just because you have a machine that doesn't have a touch screen rather identifies you as the bellend.

Personally I'm not really a huge fan of Lenovo. Had a fairly powerful i7 based laptop that was very unimpressive from a performance perspective. Currently have an HP which is better. Had a less than perfect experience with an Acer and an horrific experience with a Toshiba. So, in the laptop arena I would probably be looking at Dell and HP. Good value for money (but I can't vouch for reliability) Asus are up there. My daughter has an MSI (bought from Novatech a couple of years ago) and that has been outstanding value for money. I reckon it performed better than my work Lenovo and yet it has a weaker processor, far less memory and lower spec graphics. That said, Lenovo do make some nice looking machines at the moment but not sure I could persuade myself to buy one.

With a laptop you really are in the "get what you pay for" territory. Do you really need a laptop? Would a desktop machine suffice? If so, you will get far more bang for your buck and a wider choice of machines / vendors. Typically more easily upgradable as well. And at anything less than £500 I would say you are likely to be struggling to get a decent graphics solution on a laptop.

The recommendation of a precision above would be a good one (and I'd recommend one too, typing on one right now) but really not gonna fit in the budget!

Something's gonna have to give if £550 is the budget - i'd say ditch the discrete GPU requirement. I happily edit HD video on my 2 year old i7 MBP with integrated graphics. Producing the final render takes some time, but it'd still take appreciable time on a top end machine - you'll always be waiting.

Looking on the dell biz outlet... £500 inc vat will get you a latitude E5530 with a i5, 4Gb, 500Gb, 1080p display., win 7 pro.

Spend the change on some more ram.

Doubt you'll do much better than that for your budget, and the dell biz machines are reasonably well built.

Don't get too enticed by downgrade rights as they only apply to pro editions which most common commercial kit doesn't have.

I am no big fan of windows 8, particularly without touch like on a surface pro, but with some fettling you can get along with it fine on standard kit. Getting shot of gestures on my mouse and hot corners were my first steps to restoring sanity

Our Lenovo z510 is ace. Fast, not missed a beat, seemingly well made (if tank like) and a slightly too shiny screen -but ours is win 7.
I also have win 8 on non touch screen Acer at work, and it is great and rock steady. It is sorted to load in desktop and the keyboard shortcuts are really quick, I don't wish for a touch screen. It did take some reading of instructions and learning new habits.

If you're stuck with 8, just install ClassicShell or Start8 and turn all the useless rubbish e.g. hotspots off. Also you can boot straight to the desktop etc.
It's not a bad OS underneath all the brain-dead design decisions relating to Metrothat shitty tile screen you keep getting stuck in Modern.

The Z510 has a slightly bigger screen, but it only has an i5 processor and is £50.
They have different graphics cards though.
Not really knowledgeable enough to know which is better.
Which one would suit my needs better and why?
Cheers!